GATE Overflow - Recent questions tagged graph-coloring
http://gateoverflow.in/tag/graph-coloring
Powered by Question2AnswerTIFR2017-B-10
http://gateoverflow.in/95817/tifr2017-b-10
<p>A vertex colouring of a graph $G=(V, E)$ with $k$ coulours is a mapping $c: V \rightarrow \{1, \dots , k\}$ such that $c(u) \neq c(v)$ for every $(u, v) \in E$. Consider the following statements:</p>
<ol style="list-style-type:lower-roman">
<li>If every vertex in $G$ has degree at most $d$ then $G$ admits a vertex coulouring using $d+1$ colours.</li>
<li>Every cycle admits a vertex colouring using 2 colours</li>
<li>Every tree admits a vertex colouring using 2 colours</li>
</ol>
<p>Which of the above statements is/are TRUE? Choose from the following options:</p>
<ol style="list-style-type:upper-alpha">
<li>only i</li>
<li>only i and ii</li>
<li>only i and iii</li>
<li>only ii and iii</li>
<li>i, ii, and iii</li>
</ol>Graph Theoryhttp://gateoverflow.in/95817/tifr2017-b-10Fri, 23 Dec 2016 17:17:52 +0000TIFR2017-B-1
http://gateoverflow.in/95669/tifr2017-b-1
<p>A vertex colouring with three colours of a graph $G=(V, E)$ is a mapping $c: V \rightarrow \{R, G, B\}$ so that adjacent vertices receive distinct colours. Consider the following undirected graph.</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=12791582196260308825"></p>
<p>How many<span class="marker"> vertex colouring</span> with three colours does this graph have?</p>
<ol style="list-style-type:upper-alpha">
<li>$3^9$</li>
<li>$6^3$</li>
<li>$3 \times 2^8$</li>
<li>$27$</li>
<li>$24$</li>
</ol>Graph Theoryhttp://gateoverflow.in/95669/tifr2017-b-1Fri, 23 Dec 2016 11:39:43 +0000Algo-coloring
http://gateoverflow.in/90976/algo-coloring
Consider a tree with n nodes, where a node can be adjacent to maximum 4 other nodes.Then the minimum number of color needed to color the tree, so that no two adjacent node gets same color?Algorithmshttp://gateoverflow.in/90976/algo-coloringFri, 09 Dec 2016 11:45:05 +0000UGCNET-June2012-II-4
http://gateoverflow.in/55606/ugcnet-june2012-ii-4
<p>The number of colours required to properly colour the vertices of every planer graph is</p>
<ol style="list-style-type: upper-alpha;">
<li>2</li>
<li>3</li>
<li>4</li>
<li>5</li>
</ol>
Graph Theoryhttp://gateoverflow.in/55606/ugcnet-june2012-ii-4Mon, 04 Jul 2016 09:31:47 +0000ISRO2007-07
http://gateoverflow.in/49477/isro2007-07
<p>If a graph requires $k$ different colours for its proper colouring, then the chromatic number of the graph is</p>
<ol style="list-style-type: upper-alpha;">
<li>1</li>
<li>k</li>
<li>k-1</li>
<li>k/2</li>
</ol>
Graph Theoryhttp://gateoverflow.in/49477/isro2007-07Fri, 10 Jun 2016 06:58:12 +0000IISC-CSA-Research-Test-10
http://gateoverflow.in/49072/iisc-csa-research-test-10
A proper vertex colouring of a graph $G$ is a colouring of the vertices in $G$ in such a way that two vertices get different colours if they are adjacent. The minimum number of colours required for proper vertex colouring of $G$ is called the chromatic number of $G$. Then what is the chromatic number of the cycle graph on 149 vertices?Graph Theoryhttp://gateoverflow.in/49072/iisc-csa-research-test-10Wed, 08 Jun 2016 05:54:36 +0000CMI2015-A-04b
http://gateoverflow.in/47262/cmi2015-a-04b
<p style="line-height: 20.8px;">A college prepares its timetable by grouping courses in slots A, B, C, . . . All courses in a slot meet at the same time, and courses in different slots have disjoint timings. Course registration has been completed and the administration now knows which students are registered for each course. If the same student is registered for two courses, the courses must be assigned different slots. The administration is trying to compute the minimum number of slots required to prepare the timetable.
<br>
The administration decides to model this as a graph where the nodes are the courses and edges represent pairs of courses with an overlapping audience. In this setting, the graph theoretic question to be answered is:</p>
<p>Find a minimal colouring.</p>
Graph Theoryhttp://gateoverflow.in/47262/cmi2015-a-04bSun, 29 May 2016 12:59:34 +0000CMI2015-A-03
http://gateoverflow.in/47033/cmi2015-a-03
Suppose each edge of an undirected graph is coloured using one of three colours &mdash; red, blue or green. Consider the following property of such graphs: if any vertex is the endpoint of a red coloured edge, then it is either an endpoint of a blue coloured edge or not an endpoint of any green coloured edge. If a graph $G$ does not satisfy this property, which of the following statements about $G$ are valid?<br />
<br />
There is a red coloured edge.<br />
<br />
Any vertex that is the endpoint of a red coloured edge is also the endpoint of a green coloured edge.<br />
<br />
There is a vertex that is not an endpoint of any blue coloured edge but is an endpoint of a green coloured edge and a red coloured edge.<br />
<br />
(A) and (C).Graph Theoryhttp://gateoverflow.in/47033/cmi2015-a-03Fri, 27 May 2016 12:31:23 +0000CMI2011-B-01a
http://gateoverflow.in/46200/cmi2011-b-01a
<p>A multinational company is developing an industrial area with many buildings. They want to connect the buildings with a set of roads so that:</p>
<ul>
<li>Each road connects exactly two buildings.</li>
<li>Any two buildings are connected via a sequence of roads.</li>
<li>Omitting any road leads to at least two buildings not being connected by any sequence of roads.</li>
</ul>
<p>Is it always possible to colour each building with either red or blue so that every road connects a red and blue building?</p>
Graph Theoryhttp://gateoverflow.in/46200/cmi2011-b-01aThu, 19 May 2016 11:53:16 +0000GATE 2016-2-03
http://gateoverflow.in/39553/gate-2016-2-03
The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.Graph Theoryhttp://gateoverflow.in/39553/gate-2016-2-03Fri, 12 Feb 2016 12:02:49 +0000TIFR2013-B-1
http://gateoverflow.in/25508/tifr2013-b-1
<p>Let $G= (V, E)$ be a simple undirected graph on $n$ vertices. A colouring of $G$ is an assignment of colours to each vertex such that endpoints of every edge are given different colours. Let $\chi (G)$ denote the chromatic number of $G$, i.e. the minimum numbers of colours needed for a valid colouring of $G$. A set $B\subseteq V$ is an independent set if no pair of vertices in $B$ is connected by an edge. Let $a(G)$ be the number of vertices in a largest possible independent set in $G$. In the absence of any further information about $G$ we can conclude.</p>
<ol style="list-style-type: upper-alpha;">
<li>$\chi (G)\geq a(G)$</li>
<li>$\chi (G)\leq a(G)$</li>
<li>$a(G)\geq n/\chi (G)$</li>
<li>$a(G)\leq n/\chi (G)$</li>
<li>None of the above.</li>
</ol>Graph Theoryhttp://gateoverflow.in/25508/tifr2013-b-1Thu, 05 Nov 2015 14:47:13 +0000graph coloring
http://gateoverflow.in/11992/graph-coloring
What is the minimum number of colors needed to color a graph with five vertices. The graph contain a cycle alsoGraph Theoryhttp://gateoverflow.in/11992/graph-coloringFri, 26 Jun 2015 04:32:59 +0000GATE2006-IT-25
http://gateoverflow.in/3564/gate2006-it-25
<p>Consider the undirected graph $G$ defined as follows. The vertices of $G$ are bit strings of length $n$. We have an edge between vertex $u$ and vertex $v$ if and only if $u$ and $v$ differ in exactly one bit position (in other words, $v$ can be obtained from $u$ by flipping a single bit). The ratio of the chromatic number of $G$ to the diameter of $G$ is</p>
<ol style="list-style-type: upper-alpha;">
<li>1/(2<sup>n-1</sup>)</li>
<li>1/n</li>
<li>2/n</li>
<li>3/n</li>
</ol>Graph Theoryhttp://gateoverflow.in/3564/gate2006-it-25Fri, 31 Oct 2014 08:59:28 +0000GATE2008-IT-3
http://gateoverflow.in/3263/gate2008-it-3
<p>What is the chromatic number of the following graph?</p>
<p><img alt="GATE2008-IT_3" src="http://gateoverflow.in/?qa=blob&qa_blobid=11374265713454505408" width="320"></p>
<ol style="list-style-type:upper-alpha">
<li>2</li>
<li>3</li>
<li>4</li>
<li>5</li>
</ol>Graph Theoryhttp://gateoverflow.in/3263/gate2008-it-3Tue, 28 Oct 2014 02:17:30 +0000GATE2004-77
http://gateoverflow.in/1071/gate2004-77
<p>The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same color, is</p>
<p><img alt="" height="132" src="http://gateoverflow.in/?qa=blob&qa_blobid=224746263939656354" width="172"></p>
<ol style="list-style-type:upper-alpha">
<li>2</li>
<li>3</li>
<li>4</li>
<li>5</li>
</ol>Graph Theoryhttp://gateoverflow.in/1071/gate2004-77Fri, 19 Sep 2014 01:35:41 +0000GATE2002-1.4
http://gateoverflow.in/808/gate2002-1-4
<p>The minimum number of colours required to colour the vertices of a cycle with $n$ nodes in such a way that no two adjacent nodes have the same colour is</p>
<ol style="list-style-type:upper-alpha">
<li>$2$</li>
<li>$3$</li>
<li>$4$</li>
<li>$n-2 \left \lfloor \frac{n}{2} \right \rfloor+2$</li>
</ol>Graph Theoryhttp://gateoverflow.in/808/gate2002-1-4Mon, 15 Sep 2014 23:01:05 +0000GATE2009-2
http://gateoverflow.in/796/gate2009-2
<p>What is the chromatic number of an $n$ vertex simple connected graph which does not contain any odd length cycle? Assume $n > 2$.</p>
<ol style="list-style-type:upper-alpha">
<li>2</li>
<li>3</li>
<li>n-1 </li>
<li>n</li>
</ol>Graph Theoryhttp://gateoverflow.in/796/gate2009-2Mon, 15 Sep 2014 20:22:28 +0000